Operator Theory (MATH658) Course Detail

Course Name	Course Code	Season	Lecture Hours	Application Hours	Lab Hours	Credit	ECTS
Operator Theory	MATH658	Area Elective	3	0	0	3	5

Pre-requisite Course(s)
N/A

Course Language	English
Course Type	Elective Courses
Course Level	Natural & Applied Sciences Master's Degree
Mode of Delivery	Face To Face
Learning and Teaching Strategies	Lecture, Discussion, Question and Answer.
Course Coordinator
Course Lecturer(s)
Course Assistants
Course Objectives	This is an introductory course in Operator Theory and its applications. We will start with a short review of Banach and Hilbert Spaces and their linear bounded operators, then we turn to the Spectral Theory of linear bounded operators. Classes of Hilbert Space operators that appear frequently in applications, e.g. self adjoint, positive operators, are considered in detail. The course is aimed at Mathematics students who want to pursue a career in Analysis and its applications, but Math students from other domains and also graduate engineering students who want to understand mathematical foundations of many of the subjects considered in Engineering Mathematics are welcome as well.
Course Learning Outcomes	The students who succeeded in this course; Know the basics about spectral theory of linear bounded operators Know spectral properties of bounded self adjoint operators Know about positivity and positive operators Could apply theoretical information to concrete problems
Course Content	This is an introductory course in Operator Theory and its applications. We will start with a short review of Banach and Hilbert Spaces and their linear bounded operators, then we turn to the Spectral Theory of linear bounded operators. Classes of Hilbert Space operators that appear frequently in applications, e.g. self adjoint, positive operators, are considered in detail.

Weekly Subjects and Releated Preparation Studies

Week	Subjects	Preparation
1	A Review of Normed and Banach Spaces	[1], 2.2, 2.6, 2.7 [2], 3.1
2	Bounded operators on Normed Spaces	[1], 2.7 [2], 4.3
3	A Review of Inner Product and Hilbert Spaces	[1], 3.1,3.3,3.4 [2], 2.2—2.5
4	Hilbert Adjoint Operator	[1], 3.8,3.9
5	Spectral Theory in Normed Spaces: Introduction	[1], 7.2
6	Spectral Theory of Bounded Linear Operators	[1], 7.3
7	Spectral Mapping Theorem	[1], 7.4
8	Review and Midterm
9	Spectral Theory of Bounded Self Adjoint Linear Operators	[1], 9.1
10	Spectral Theory of Bounded Self Adjoint Linear Operators	[1], 9.2
11	Positive Operators	[1], 9.3
12	Square Roots of a Positive Operator	[1], 9.4
13	Projection Operators	[1], 9.5
14	Further Properties of Projections	[1], 9.6
15	Further properties of Projections	[1], 9.6
16	Review

Sources

Course Book	1. E. Kreyszig, Introductory Functional Analysis with Applications, Wiley Clas. Lib. Ed, 1989.
	2. G. Chacón, H. Rafeiro, J. Vallejo, Functional Analysis, De Gruyter, 2017.

Evaluation System

Requirements	Number	Percentage of Grade
Attendance/Participation	-	-
Laboratory	-	-
Application	-	-
Field Work	-	-
Special Course Internship	-	-
Quizzes/Studio Critics	-	-
Homework Assignments	3	30
Presentation	1	20
Project	-	-
Report	-	-
Seminar	-	-
Midterms Exams/Midterms Jury	1	20
Final Exam/Final Jury	1	30
Toplam	6	100

Percentage of Semester Work	70
Percentage of Final Work	30
Total	100

Course Category

Core Courses	X
Major Area Courses
Supportive Courses
Media and Managment Skills Courses
Transferable Skill Courses

The Relation Between Course Learning Competencies and Program Qualifications

#	Program Qualifications / Competencies	Level of Contribution
#	Program Qualifications / Competencies	1	2	3	4	5
1	Has the ability to apply scientific knowledge gained in the undergraduate education and to expand and extend knowledge in the same or in a different area.
2	Has the ability to obtain, to evaluate, to interpret and to apply information by doing scientific research.
3	Can apply gained knowledge and problem solving abilities in inter-disciplinary research.
4	Has the ability to work independently within research area, to state the problem, to develop solution techniques, to solve the problem, to evaluate the obtained results and to apply them when necessary.
5	Takes responsibility individually and as a team member to improve systematic approaches to produce solutions in unexpected complicated situations related to the area of study.
6	Can develop strategies, implement plans and principles on the area of study and can evaluate obtained results within the framework.
7	Can develop and extend the knowledge in the area and to use them with scientific, social and ethical responsibility.
8	Has the ability to follow recent developments within the area of research, to support research with scientific arguments and data, to communicate the information on the area of expertise in a systematically by means of written report and oral/visual presentation.
9	Has proficiency in English language and has the ability to communicate with colleagues and to follow the innovations in mathematics and related fields.
10	Has software and hardware knowledge in the area of expertise, and has proficient information and communication technology knowledge.
11	Has professional ethical consciousness and responsibility which takes into account the universal and social dimensions in the process of data collection, interpretation, implementation and declaration of results in mathematics and its applications.

ECTS/Workload Table

Activities	Number	Duration (Hours)	Total Workload
Course Hours (Including Exam Week: 16 x Total Hours)	16	3	48
Laboratory
Application
Special Course Internship
Field Work
Study Hours Out of Class	14	2	28
Presentation/Seminar Prepration	1	10	10
Project
Report
Homework Assignments	3	5	15
Quizzes/Studio Critics
Prepration of Midterm Exams/Midterm Jury	1	12	12
Prepration of Final Exams/Final Jury	1	12	12
Total Workload			125